Day 1 Session 3 Key concepts of stock assessment modelling

Day 1 Session 3 Key concepts of stock assessment modelling
12/4/2018 Day 1 Session 3 Key concepts of stock assessment modelling

Overview of key concepts
12/4/2018 Overview of key concepts Fish stocks: What are they? What is stock assessment? What is a stock assessment model? How does a stock assessment model “work”? Mathematical component (and a note about equations) Statistical component (and an introduction to abundance indices and model fitting) Types of model WCPO tuna models Other models

Concept 1: The fish stock
12/4/2018 Concept 1: The fish stock Key Concept 1 A stock assessment model is used to assess a fish population that has little or no mixing or interbreeding with other populations. Definition of a “stock” “A unit stock is an arbitrary collection [of a single species] of fish that is large enough to be essentially self reproducing (abundance changes are not dominated by immigration and emigration) with members of the collection showing similar patterns of growth*, migration and dispersal. The unit should not be so large as to contain many genetically distinct races of subpopulations within it.” Hilborn and Walters (1991)

12/4/2018 Concept 1: The fish stock Why do we manage and assess fisheries at the level of a stock? 1. Little or no external influences, self contained 2. Scientifically meaningful 3. Management convenience Stock

12/4/2018 Concept 1: The fish stock How do we identify a fish stock? It’s a very difficult task ………often little clear information. We can use: Genetics Tagging CPUE analyses Morphometrics Often stock assessments are conducted on “stocks” where there is some uncertainty regarding the boundaries of the stock (e.g. WCPO v EPO bet/yft; SWPO v SPO v PO stm)

12/4/2018 Concept 1: The fish stock Tuna stocks in the WCPO Yellowfin tuna Limited mixing and genetic variation found Bigeye tuna Mixing less limited and no genetic variation found Schaefer, K.M. and D.W. Fuller (2009). Horizontal movements of bigeye tuna (Thunnus obesus) in the Eastern Pacific Ocean, as determined from conventional and archival tagging experiments initiated during IATTC Bull. 24(2):

12/4/2018 Concept 1: The fish stock Tuna stocks in the WCPO Albacore tuna Low catch and CPUE in equatorial waters suggests limited adult mixing No tag exchange between north and south PO tagged fish Discrete spawning areas (based on larval surveys) between North and south Pacific

12/4/2018 Concept 1: The fish stock Tuna stocks in the WCPO: Skipjack tuna There is uncertainty regarding skipjack stock structure in the Pacific, but given a lack of evidence for trans basin movements and generally localised tag returns in general, mixing is thought to be limited within generations (short lived species) and in the medium term, meaning the WCPO “stock” is assessed as such for management purposes.

12/4/2018 Concept 1: The fish stock Striped marlin Equatorial CPUEs low, genetic variation between SWPO and NEPO, discrete spawning areas?, no transbasin tag returns, no transbasin movement indicated by 50 PSAT tags, all indicate limited mixing and a potential southwest Pacific stock (for management purposes) Bromhead et al (2004) Courteousy Michael Domeier, Pfelger Institute, 2006 Bromhead et al (2004)

Concept 2: The stock assessment process
12/4/2018 Concept 2: The stock assessment process Stock assessment is a multi-step process that starts with management questions, and includes processes involved in data collection, model selection, stock assessment modelling, and subsequent advice to decision makers. **Group discussion

Concept 3: The stock assessment model
12/4/2018 Concept 3: The stock assessment model A stock assessment model provides a mathematical simplification of a very complex system (fish and fishery), to help us estimate population changes over time in response to fishing

12/4/2018 Concept 3: The stock assessment model Overview: In explaining to you what a stock assessment model is, we are going to discuss the following: What is an equation? (because stock assessment models comprise lots of them) What is a [mathematical] model? What is a stock assessment model? How does a stock assessment model “work”? Mathematical component Statistical component (and an introduction to abundance indices and model fitting) 5. What are the different types of stock assessment model?

12/4/2018 Concept 3: The stock assessment model 1. Equations An equation can be thought of as simply being a sentence, but the words have been replaced by symbols….for example: Bt+1=Bt+R+G-M-C If we know what the symbols mean, we can read the sentence! “Population biomass next year is equal to the biomass this year, plus biomass of new recruits in one years time, plus biomass of additional growth of this years fish, minus the biomass of fish that died of natural causes, minus the biomass of fish killed by fishing”.

12/4/2018 Concept 3: The stock assessment model 1. Equations Large stock assessment models can involve complex equations expressing mathematical and statistical functions which attempt to describe the interacting fishery and fish population processes. Interpreting those interlinking equations requires training in maths, statistics and computer programming… However, we don’t need to have all that training to get a basic understanding of how stock assessment models work…

12/4/2018 Concept 3: The stock assessment model 1. Equations We will focus on a very basic model of an exploited fish population during this workshop, which uses very simple equations to encorporate nearly all the major elements that typically comprise far more complex assessments, such as those conducted for tuna with MULTIFAN-CL in the Western and Central Pacific. If you can gain an understanding of how our workshop model works, you will be a long way towards understanding the key principles and mechanics that underpin the Western and Central Pacific tuna assessment models. Nt+1,a+1 = Nt,ae-(Ma + Ft,a) Ft,a = qtEtsa Ct,a = Nt,aFt,awa Rt = (ASt)/(b+St) Nt+1,1 = Rt Bt = ΣNt,awa St = ΣNt,awaoa VBt = ΣNt,awasa

12/4/2018 Concept 3: The stock assessment model 2. What is a [mathematical] model? A mathematical representation (or description) of a system or process that is used to help us understand the system and how the system works Lets look at a really simple example of a model, in this case a model comprising only one equation: This simple single equation model is a mathematical description of a biological process (growth!). It describes how quickly the animal gets bigger as it gets older. Easy! (NB: But a critical question we will address later is… how do we know if our model is “true” or realistic of the process it is trying to describe? ….we’ll come back to this) Of course, if stock assessment models were this simple, we wouldn’t need to be here!

12/4/2018 Concept 3: The stock assessment model 3. What is a stock assessment model? A stock assessment model provides a mathematical simplification of a very complex system (the fish population and fishery), to help us estimate population changes over time in response to fishing. They serve as a tool to assist the provision of scientific advice to fisheries managers and policy makers in relation to the impact of fishing upon the status (health) of the stock (past, current, future), at the same time taking into consideration other factors influencing stock abundance (environmental impacts on recruitment etc). Stock assessment models can be used to make predictions regarding the response of the stock to different management actions. There are many different types of assessment model. In this workshop, we will concentrate on age structured models. This is what a stock assesssment model is in general terms. (purpose etc). Its actual structure and mechanism is explained in the following sections.

Stock Assessment Models
12/4/2018 Stock Assessment Models 4. How does a stock assessment model work? A stock assessment model can be considered to comprise two key components, these being: A mathematical model of population processes A statistical model used to fit the mathematical model to data collected from the fishery

12/4/2018 Stock Assessment Models 4. How does a stock assessment model work? 1. Mathematical model of the exploited fish population dynamics: To estimate abundance (biomass) over time, the model must take into account (at the very least) four key processes: Recruitment, Growth, Natural Mortality and Fishing Mortality, conceptually expressed as: Bt+1=Bt+R+G-M-C Growth (G) Recruitment (R) Natural mortality (M ) Fishing mortality (F ) Biomass Biomass added Biomass removed Remember, all a SAM is trying to do is account for the interaction of four processes. Whenever you get confused or overwhelmed by information this week, just remember that at its core is a very simple principle…..balancing biomass between these processes. Of course, this is just a conceptual equation and diagram. An actual assessment model looks much more complex….as follows…

12/4/2018 Stock Assessment Models 4. How does a stock assessment model work: An example of an age structured model: Nt+1,a+1 = Nt,ae-(Ma + Ft,a) Ft,a = qtEtsa Ct,a = Nt,aFt,awa Rt = (ASt)/(b+St) Nt+1,1 = Rt Bt = ΣNt,awa St = ΣNt,awaoa VBt = ΣNt,awasa Nt+1,a+1 = Number of fish of age+1 at time+1 Ma = natural mortality rate at age a Fa = fishing mortality rate at age a q = catchability E = fishing effort (units) s = age specific vulnerability to the gear (selectivity of the gear) Ct,a = Catch at time t and age a wa = Mean weight at age a << (Growth) Rt = Recruitment at time t A = maximum recruitment b = Stock size when recruitment is half the maximum recruitment wa = weight at age a oa = proportion mature at age a Bt = population biomass at time t St = spawning stock biomass at time t VB = vulnerable biomass at time t Here we can see a simple age structure model (the mathematical component describing the interaction between the four key processes). Note that each of the key processeses is highlighted in red

12/4/2018 Stock Assessment Models How does a stock assessment model work: “Counting fish is just like counting trees…except we cant see them and they move!!” The mathematical component of the model comprises equations describing each of these processes and how they interact with each other to determine the population biomass (and other parameters) over time. We build a model because we cant directly “count” the exact numbers of recruits and deaths nor measure the growth of each fish in the population … …instead of direct counts and measures, our model (via a series of equations) allows us to “estimate” these processes and the populations dynamics. Nt+1,a+1 = Nt,ae-(Ma + Ft,a) Ft,a = qtEtsa Ct,a = Nt,aFt,awa Rt = (ASt)/(b+St) Nt+1,1 = Rt Bt = ΣNt,awa St = ΣNt,awaoa VBt = ΣNt,awasa Read from slide…..

12/4/2018 Stock Assessment Models 4. How does a stock assessment model work (cont’d)? Nt+1,a+1 = Nt,ae-(Ma + Ft,a) Ft,a = qtEtsa Ct,a = Nt,aFt,awa Rt = (ASt)/(b+St) Nt+1,1 = Rt Bt = ΣNt,awa St = ΣNt,awaoa VBt = ΣNt,awasa These equations are of little use unless we have data or information which can accurately inform the model of the value of most of the key parameters: How much fishing effort? How much catch? Whats the average weight of fish in each age class? What proportion of the fish in each age class are mature? Etc, etc We have to collect the data required to inform the model regarding the value of of these parameters

12/4/2018 Stock Assessment Models 4. How does a stock assessment model work? (cont’d) Nt+1,a+1 = Nt,ae-(Ma + Ft,a) Ft,a = qtEtsa Ct,a = Nt,aFt,awa Rt = (ASt)/(b+St) Nt+1,1 = Rt Bt = ΣNt,awa St = ΣNt,awaoa VBt = ΣNt,awasa However, the reality is that we have variable levels of information or data pertaining to the different parameters: Some parameters we have very good estimates for (e.g. maybe catch, average size at age) derived from biological research or fishery data collection Some parameters we have limited data for and some uncertainty Some parameters we have no data for and high uncertainty (unknown parameter values)

12/4/2018 Stock Assessment Models 4. How does a stock assessment model work? (cont’d) Nt+1,a+1 = Nt,ae-(Ma + Ft,a) Ft,a = qtEtsa Ct,a = Nt,aFt,awa Rt = (ASt)/(b+St) Nt+1,1 = Rt Bt = ΣNt,awa St = ΣNt,awaoa VBt = ΣNt,awasa Given that we have moderate or large uncertainty regarding the value of some parameters, how do we determine their value in an assessment model? How do we determine that the model overall can accurately predict the population dynamics and status of the fish population? (i.e. is an accurate model of “reality”) We do this by a process called “fitting” the model to fishery data using the second component of the model, the statistical component. How does this work?

Model fitting How does model fitting work?
12/4/2018 Model fitting How does model fitting work? We need an index of abundance To make sure our model can accurately predict how the population size changes over time, we need to collect and provide the model with data from the fishery itself which acts as an indicator of those changes in population size: i.e; data which can act as an index of abundance (or an index of relative population size ) over time. Typically the index used is catch rate or catch per unit effort (CPUE) data, generally using data collected from fishers logsheets: CPUE = catch/effort (e.g. 6 fish/1000 hooks; 2mt/set)

12/4/2018 Model fitting How does model fitting work? 2. How is CPUE data an “index of abundance”? The use of CPUE relies on the assumption that the relationship between the index (CPUE) and abundance is linear (proportional), so if CPUE goes up, the population has gotten bigger; if it goes down, it has gotten smaller. In this way CPUE is assumed to be an accurate index of population change over time. (**In fact, this is not always true, but we will discuss this further later) Abundance (Biomass) Abundance Index Time Index Biomass

12/4/2018 Model fitting How does model fitting work? 2. How is CPUE data an “index of abundance”? pond with 12 fish pond with 24 fish Mean CPUE ~ 1.2/set Mean CPUE ~ 2.4/set Note that fish must be evenly distributed.

Model fitting How does “model fitting” work?
12/4/2018 Model fitting How does “model fitting” work? So we have our observed CPUE series Our model also has an equation to predict CPUE. We can now employ the statistical component of the model to search for and select the “best” combination of parameter values (for the “unknown” parameters) which maximises the models ability to accurately predict the observed CPUE data (i.e.; pick the values which maximise the fit of the model predictions to the observed data). Note that the tuna assessments also fit to other data types: Tagging data (to ensure realistic modelling of movement) Size data (to ensure realistic modelling of population structure) Or use in any other stock assessment for that matter

Model fitting What are the different approaches to model fitting?
12/4/2018 Model fitting What are the different approaches to model fitting? There are (at least) three general approaches to how we might go about doing this that you should be aware of (or at least know exist!): Least-squares estimation Maximum likelihood estimation Bayesian estimation We use maximum-likelihood estimation (MLE) In our tropical tuna assessments, method (ii) (MLE) is most commonly used to fit our assessment models to our data, with that data typically being the CPUE data (an index of population size), the size data (an index of population structure) and the tagging data (an index of population movement) Or use in any other stock assessment for that matter

12/4/2018 Model fitting There are two methods we will discuss today by which stock assessment scientists fit models to observation data are: Minimisation of Sums of Squares of Errors Basically, this approach asks “What combination of values result in there being the smallest difference (degree of error) between the model estimated CPUE series and the real CPUE series?” Maximum likelihood approach This approach asks “What combination of values for all of these parameters would most likely result in the observed CPUE values occuring?” SSE and ML serve as criterion by which to judge the quality of the fit

Model fitting: MSSE approach
12/4/2018 Model fitting: MSSE approach 1. Minimisation of Sums of Squares of Errors This approach involves a search for the parameter values which minimise the sums of squared differences between the observed data and the data as predicted by the model and parameters. It is almost impossible in any slightly complex system to create a model that exactly fits the real data….there is always some error. The objective of the SSE approach is to find parameter values that minimise the total error.

Which is correct (or best reflects the actual relationship)???
12/4/2018 Model fitting: MSSE approach An example using fish length and weight: y=mx y=2x y=1x Which is correct (or best reflects the actual relationship)??? Weight (g) y=0.5x Let look at model fitting using a very simple example: Imagine we are interested in understanding the relationship between fish length and weight. Now we already have a pretty good idea of what the general relationship will be between weight and length…..who here can tell me what that is….as the fish grows in length we would expect weight to ……….increase or decrease (?). Increase, exactly. So we choose a simple model that reflects this understanding: y=mx ……or………weight = m x Length. Note that real length weight relationships are more complex than this but we are keeping it very simple to demonstrate the fitting concept. The “m” reflects the fact that we don’t know exactly how much weight will increase for each unit increase in length (i.e. we don’t know the value of “m”). For example if we plot y=1x (or y=x) then it looks like this…for every unit increase in length there is one unit increase in weight. At 1cm length the fish is 1 gram, at 5cm it is 5 grams et etc But we don’t know m, it could be any value ….. Does weight increase by 0.5grams for every 1 cm increase in weight? (show line) ……..by 2.0 grams?(show line) We could choose one of these values but how would we know it reflected the true relationship for actual fish in the sea? We need to go and collect some data from actual fish, and use that data to guide us in calculating what the value of “m” is and ensuring the overall model (y=mx) is an accurate reflection of the actual relationship between length and weight. So heres our data. The first thing you notice is that the data is not all in a straight line but scattered. This is because, just like humans, the fish will vary somewhat in the exact relationship……some fish of a particular length will weight slightly more or less than other fish of the same length. But its also clear that our assumed increasing relationship is overall, correct. So now we want to know two things: We now want to “fit” our model to the observed data, and determine the “average relationship” ……e.g. what is the mean weight of a fish if the mean length is x. To do this, we get a computer programme so search through every possible value of m, our unknown parameter, until it finds the value which provides the line (and model) with the best fit. How does the computer know which line has the best fit? predict the model fit line One way to fit models (via SSE) We can specify an equation for x and y based on our prior understanding that when x increases, so does why. But we arnt sure how much y increases when x does. The m in y=mx is unknown. So it doesn’t matter, we go out and collect some data and here it is. Certainly it seems to be an positive relationship. We then go and use a computer to work out exactly how much y increases (on average) for each unit increase in x. The computer uses maths to “fit” the model (y=mx) to the observed data, by searching through all the possible values of m until it finds the one that gives the closest fit. We will have a more detailed session on model fitting later in the workshop – but if you can understand this then you have an understanding of the basic process and how it applies in stock assessment. (could develop this into a very simple graph sheet exercise – yes, do this – give them the equation, and 5 “observed” dots – and the SSE formula. 5 4 3 2 1 Length (cm)

12/4/2018 Model fitting: MSSE approach An example using fish length and weight: y=mx y=2x y=1x Observed values Weight (g) y=0.5x So heres our data. The first thing you notice is that the data is not all in a straight line but scattered. This is because, just like humans, the fish will vary somewhat in the exact relationship……some fish of a particular length will weight slightly more or less than other fish of the same length. But its also clear that our assumed increasing relationship is overall, correct. So we want to find the model that best describes the relationship between length and weight in actual worms….how do we do this. Well, we get a computer programme to search through every possible value of m, our unknown parameter, until it finds the value which provides the line (and model) that best describes the relationship between length and weight, as indicated by our observed data. We can guess by eye which model most accurately describes the relationship….as follows: [click up green line] ……do you think that this model describes the relationship well? Could we use it to predict fish weight if we only knew the length? [click up blue line] ……do you think that this model describes the relationship well? Could we use it to predict fish weight if we only knew the length? [click up red line] ……do you think that this model describes the relationship well? Could we use it to predict fish weight if we only knew the length? How does the computer know which line has the best fit? predict the model fit line One way to fit models (via SSE) We can specify an equation for x and y based on our prior understanding that when x increases, so does why. But we arnt sure how much y increases when x does. The m in y=mx is unknown. So it doesn’t matter, we go out and collect some data and here it is. Certainly it seems to be an positive relationship. We then go and use a computer to work out exactly how much y increases (on average) for each unit increase in x. The computer uses maths to “fit” the model (y=mx) to the observed data, by searching through all the possible values of m until it finds the one that gives the closest fit. We will have a more detailed session on model fitting later in the workshop – but if you can understand this then you have an understanding of the basic process and how it applies in stock assessment. (could develop this into a very simple graph sheet exercise – yes, do this – give them the equation, and 5 “observed” dots – and the SSE formula. 5 4 3 2 1 Length (cm)

12/4/2018 Model fitting: MSSE approach Models used to deduce relationship and find best fit SSE = Sum (Observed-Predicted)2 Observed values y=0.5x Predicted values Difference between the observed and predicted value is the “residual error”

12/4/2018 Model fitting: MSSE approach Models used to deduce relationship and find best fit Predicted values SSE = Sum (Observed-Predicted)2 Observed values KEY STATEMENT – on average, the distance between the observed and predicted values is much less for this model than for the other model. Therefore this model must be a more realistic model (i.e. it is a better “fit”) than the other model. We’d have much higher confidence to use this model to predict lengths from weights or weights from lengths than we would the other model. Now this is a very simple example of the process. But in effect, stock assessment model fitting is not different in principle….lets look again at our previous simple assessment models……. Difference between the observed and predicted value is the “residual error”

12/4/2018 Model fitting: MSSE approach An example of an age structured model: Nt+1,a+1 = Nt,ae-(Ma + Ft,a) Ft,a = qtEtsa Ct,a = Nt,aFt,awa Rt = (ASt)/(b+St) Nt+1,1 = Rt Bt = ΣNt,awa St = ΣNt,awaoa VBt = ΣNt,awasa Nt+1,a+1 = Number of fish of age+1 at time+1 Ma = natural mortality rate at age a Fa = fishing mortality rate at age a q = catchability E = fishing effort (units) s = age specific vulnerability to the gear (selectivity of the gear) Ct,a = Catch at time t and age a wa = Mean weight at age a << (Growth) Rt = Recruitment at time t A = maximum recruitment b = Stock size when recruitment is half the maximum recruitment wa = weight at age a oa = proportion mature at age a Bt = population biomass at time t St = spawning stock biomass at time t VB = vulnerable biomass at time t

12/4/2018 Model fitting: MSSE approach Model fitting…. Here are the equations that described the population processes….we believe these are correct based on past research etc. And we have data collected from the fishery or science research for many of the parameters. But we don’t have any data for some parameters. Model fitting is the process by which our computer programme searches amongst all the possible “unknown” parameter values until it finds and selects the “best” combination of parameter values which maximise the fit of the model predictions to the observed data (e.g. CPUE). In other words it selects values for the unknown parameters which maximise the models ability to accurately predict the observed CPUE (in this example) data. Nt+1,a+1 = Nt,ae-(Ma + Ft,a) Ft,a = qtEtsa Ct,a = Nt,aFt,awa Rt = (ASt)/(b+St) Nt+1,1 = Rt Bt = ΣNt,awa St = ΣNt,awaoa VBt = ΣNt,awasa

Model fitting A note on parameter estimation
12/4/2018 Model fitting A note on parameter estimation However, we don’t just allow the model to pick ANY value for the uncertain or unknown parameters however! Typically, we provide the model: A start value (from which it begins its search) Some boundary or limit values (to which it must constrain its search within) A likelihood distribution (penalty) which specifies to the model which values within the range specified will be more likely. Likelihood profile (penalty) Or use in any other stock assessment for that matter Lower limit Start value Upper limit

Model fitting: ML approach
12/4/2018 Model fitting: ML approach 2. The Maximum Likelihood Method Parameters are selected which maximise the probability or likelihood that the observed values (the data) would have occurred given the particular model and the set of parameters selected (the hypothesis being tested). The set of parameter values which generate the largest likelihood are the maximum likelihood estimates: So.. Likelihood = P{data|hypothesis} Which means “the probability of the data (the observed values) given the hypothesis (the model plus the parameter values selected)”. E.g. Think of the flip of a coin. Whats the probability of getting heads? Of getting tails? Stock assessment models can use fairly complex statistics to determine the probability of, for example, the observed CPUE series occuring, given a particular model.

Model fitting: Summary
12/4/2018 Model fitting: Summary We use our knowledge of population processes to build a mathematical model of the population that has equations to describe all the processes, how they link together, and how they influence population size over time. Each equation will be made up of different components (or parameters) Some of the parameter values we will know already (e.g from biological research, from fisheries catch effort data collection, etc). Some of the parameters will have unknown values. We use a statistical model (and computer) to go through all the different combinations of possible values for those unknown parameters, until it finds a combination that allows the model to accurately predict the observed CPUE. In other words, produce a CPUE time series that fits or matches (i.e. differs very little from..) the real CPUE time series,which we believe is an accurate index of changes in population size over time. ** This description describes some of the core principles only

Model types and selection
12/4/2018 Model types and selection There are many different types of fish stock assessment model that can be used and selecting an appropriate model is dependant on the management question being asked and the data that is available. What are the various types of stock assessment model? How do they differ? Age structure Fishermen dynamics Biomass Dynamics Models Ecosystems and multispecies models Spatial models Modified from Hillborn and Walters, 1992 WE ARE GOING TO FOCUS ON AGE STRUCTURED MODELS IN THIS WORKSHOP

Additional useful information
12/4/2018 Additional useful information

Stock assessment involves equations
12/4/2018 Stock assessment involves equations To understand how stock assessments work at a technical level, it is important to understand the key types of equation used and how they are used…. …..the following section provides a very brief and basic overview of the use of equations in stock assessment modeling

12/4/2018 Stock assessment involves equations A quick word about equations and model building! An equation can be thought of as simply being a sentence, but the words have been replaced by symbols….for example: Bt+1=Bt+R+G-M-C If we know what the symbols mean, we can read the sentence! “Population biomass next year is equal to the biomass this year, plus biomass of new recruits in one years time, plus biomass of additional growth of this years fish, minus biomass of fish that died of natural causes, minus the biomass of fish killed by fishing”. Large stock assessment models can involve complex equations expressing mathematical and statistical functions which attempt to describe the interacting fishery and fish population processes. Interpreting those interlinking equations requires training in maths, statistics and computer programming…

12/4/2018 Stock assessment involves equations A quick word about equations and model building! However, we don’t need to have all that training to get a basic understanding of how stock assessment models work We will focus on a very basic model of an exploited fish population during this workshop, which incorporates nearly all the major elements that comprise far more complex assessments, such as those conducted for tuna with MULTIFAN-CL in the Western and Central Pacific. If you can gain an understanding of how our workshop model works, you will be a long way towards understanding the key principles and mechanics that underpin the Pacific tuna assessment models.

12/4/2018 Stock assessment involves equations The equations making up our simple stock assessment model Nt+1,a+1 = Nt,ae-(Ma + Ft,a) Ft,a = qtEtsa Ct,a = Nt,aFt,awa Rt = (ASt)/(b+St) Nt+1,1 = Rt Bt = ΣNt,awa St = ΣNt,awaoa VBt = ΣNt,awasa Nt+1,a+1 = Number of fish of age+1 at time+1 Ma = natural mortality rate at age a Fa = fishing mortality rate at age a q = catchability E = fishing effort (units) s = age specific vulnerability to the gear (selectivity of the gear) Ct,a = Catch at time t and age a wa = Mean weight at age a << (Growth) Rt = Recruitment at time t A = maximum recruitment b = Stock size when recruitment is half the maximum recruitment wa = weight at age a oa = proportion mature at age a Bt = population biomass at time t St = spawning stock biomass at time t VB = vulnerable biomass at time t

12/4/2018 Stock assessment involves equations Two of the key types of equation used in stock assessment models are: Differential equations – measure rates of change Difference equations – predict values at fixed point in time If you would like to learn more about these types of equation and why they are important in stock assessment models, please refer to the following 8 slides…… ….the key point is that the estimation of each of the 4 key processes requires equations, and the full suite of equations together make up the assessment model…

12/4/2018 Stock assessment involves equations Remember this equation? Hypothetically it might be used in an assessment model to describe how the fish grow in size as they get older. This is really all any of the equations in the model are doing, describing processes in the population…the population dynamics including its interation with the fishery (NB: But again….a critical question we will address later is… how do we know if our model is “true” or realistic of the process it is trying to describe? ….we’ll come back to this) In fact this equation is not realistic of most fish growth relationships but it serves to illustrate the point we want to make.

12/4/2018 Stock assessment involves equations Calculating rates (of change) ..we do it every day.. Car speed (km/hour) Interest rates Weight loss (kg/week) Typing (words/minute) What is a rate? It is the extent to which a change in one quantity affects a change in another related quantity. This is called a rate of change. 70 60 50 Distance (km) 40 30 20 10 Time (hours)

12/4/2018 Stock assessment involves equations Example Two cars, A and B Car A travels 80km after 4 hours Car B travels 40km after 8 hours How do we calculate the rate of travel? Rate = change in y change in x Car A: Rate = 80/ = 20km/hour Car B: Rate = 40/8 = 5km/hour Car A So…. Car A is travelling at a faster rate than Car B 70 60 50 Distance (km) Car B 40 30 20 10 Time (hours)

12/4/2018 Stock assessment involves equations Relevance to stock assessment? e.g. fish growth rates Growth rate impacts population size Natural mortality Size at maturity Biomass increase in existing pop Therefore estimating growth rates is important in predicting population change over time This is just one example of how rates may be used in stock assessment Species A So…. Species A grows at a faster rate than species B 70 60 50 Length (cm) Species B 40 30 20 10 Time (years)

12/4/2018 Stock assessment involves equations Relevance to stock assessment? Often the rates are not constant as in this example e.g. fish growth rates (in length) rates slow as the fish get older e.g. fish survival rates may increase as fish get older and then decrease when very old. Species A 70 60 50 Species B Length (cm) 40 30 20 10 Time (years)

12/4/2018 Stock assessment involves equations What types of equation are used in stock assessment models? Differential equations – measure rates of change Difference equations – predict values at fixed point in time

12/4/2018 Stock assessment involves equations Difference Equations Predicting values at fixed points in time. Example: Two populations, A and B. Population A grows at fish per year. Population B grows at 5000 fish per year. How do we calculate the population size 4 years into the future? Rate = change in y/change in x But we want to know y! Species A 20000=y/4 4*20000=y 80000=y Species B 5000=y/4 4*5000=y 20000=y Species A So…. Population A grows at a faster rate than Population B 70 60 50 Species B Population size (x1000) 40 30 20 10 Time (years)

Key Concept 5: Stock Assessment involves equations!
12/4/2018 Key Concept 5: Stock Assessment involves equations! Equations which estimate a value at a fixed point (e.g. in time) are called difference equations We already know one very well! Bt+1=Bt+R+G-M-C Many stock assessments these days are based on difference equations….they are easier to understand and more intuitively logical However, each of the components of such an equation may require estimation by another equation, and often these equations can involve the calculation of rates (ie. Use of differential equations)

Key Concept 5: Stock Assessment involves equations!
12/4/2018 Key Concept 5: Stock Assessment involves equations! For example, a basic logistic growth model We can write it to calculate change in biomass over time: dB/dt = rB(1-Bt/k)-C Or we can write it to predict biomass at some time in the future Bt+1 = Bt + rB(1-Bt/k)-Ct

Model fitting: ML approach
12/4/2018 Model fitting: ML approach Maximum Likelihood: An example Lets look at our previous example used to describe SSE approach but this time apply a Maximum likelihood approach. We have a model for the relationship between two parameters, y=mx+c x and c are known, m is not We can specify a starting mean value and probability distribution around the mean which provides the model an indication of the most likely value but some freedom to estimate alternate values if they provide a better fit. The model searches through all possible values of m until it finds a value which maximises the likelihood of the observed data, given the model specified (including the specified probability distribution) Model: y = m x x +c Where a is unknown Mean y = 1 x x +c Good fit The model searches all values of “a” until it finds the one that provides the best fit – in this case as determined by minimising the SSE. Observed values Parameter Y y = 0.5 x x +c Poor fit Parameter X

Model fitting - Maximum likelihood
12/4/2018 Parameter Y y = 1 x x +c Good fit y = 0.5 x x +c Poor fit Parameter X The model searches all values of “a” until it finds the one that provides the best fit – in this case as determined by minimising the SSE.

12/4/2018 y = 1 x x +c Good fit y = 0.5 x x +c Poor fit The model searches all values of “a” until it finds the one that provides the best fit – in this case as determined by minimising the SSE.

12/4/2018 These represent the models expected probability distribution. These are then compared to the actual distribution of the observations around the predicted relationship The model searches all values of “a” until it finds the one that provides the best fit – in this case as determined by minimising the SSE.

12/4/2018 The model searches all values of “a” until it finds the one that provides the best fit – in this case as determined by minimising the SSE.

12/4/2018 Models expected probability distribution DOES NOT fit well with the actual distribution of the fishery data Models expected probability distribution fits well with the actual distribution of the fishery data The model searches all values of “a” until it finds the one that provides the best fit – in this case as determined by minimising the SSE.

12/4/2018 The likelihood of the observed data, given the red model (left), is much higher than the likelihood of the observed data occuring given the green model (right), and the red model is chosen as the model of best fit (or most likely model). In the stock assessments, there are many unknown parameters which the software will “search” through the possible values of until it finds a model with a combination of estimated values which maximise the likelihood of the observed data occuring The model searches all values of “a” until it finds the one that provides the best fit – in this case as determined by minimising the SSE.

Maximum-likelihood estimation in brief
Note! Typically, to find the MLEs we actually search for parameter values that minimise the model’s negative log-likelihood function, logL, So, in an assessment using MLE, to determine the model with the best fit, find the model with the lowest negative log-likelihood score. A table is usually produced which provides the negative log-likelihood estimates for each data set offered to the model (e.g., catch, size, tagging data) and the combined total for all of these.

Day 1 Session 3 Key concepts of stock assessment modelling

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Day 1 Session 3 Key concepts of stock assessment modelling

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